Exploring New Frontiers in Geometry:
in the World Around Us and in Our Classrooms

A full-day symposium devoted to "Exploring New Frontiers in Geometry: in the World Around Us and in Our Classrooms", organized by Colm Mulcahy, Spelman College, and David Henderson, Cornell University, will be held on Friday, Feb 13, 1998, as part of the AAAS (American Association for the Advancement of Science) 150th anniversary celebration Annual Meeting and Science Innovation Exposition in Philadelphia, Feb 12-17, 1998.

During the two-hour break between the morning and afternoon geometry sessions, there will be a special Topical Lecture on "How Geometry is Changing Hollywood", delivered by Dr. Tony DeRose of Pixar Animation Studios. Dr. DeRose: "Did you ever wonder how the liquid metal terminator effect in {\em Terminator 2} was done? Or how {\em Toy Story} was made? This answer is that sophisticated geometric techniques were used, in conjunction with talented art direction and computer technology. This talk will provide an accessible behind-the-scenes glimpse into digitial movie making, paying particular attention to the fundamental role of geometry." Pixar Animation Studios has been a pioneer in computer animation for many years, first coming to the public's attention with the groundbreaking and Academy Award winning video {\em Tin Toy} in 1987, and then collaborating with Walt Disney Studios for the world's first full length computer animated feature {\em Toy Story} in 1995. Dr. DeRose spent a decade as a leader in computer graphics at the University of Washington before his recent move to Pixar. His expertise encompasses both computer aided geometric design (CAGD) and wavelets.

In the three-hour morning session of the geometry symposium, ``Geometry is Alive!'', the continually-evolving nature of geometry will be highlighted, together with some cutting-edge applications to, and interactions with, science and technology. Extensive use will be made of video as an aid in understanding and discovery.

Carolyn Gordon, Dartmouth College, will start the ball rolling with "Can You Hear the Shape of a Drum?"; Prof. Gordon comments, "In spectroscopy, one attempts to recover information about an object such as its shape or chemical decomposition from the frequency spectrum of light or sound the object emits. Recently David Webb and Scott Wolpert and I discovered that the answer to Mark Kac's 1964 question, "Can one hear the shape of a drum?" is negative. In other words, there are drums with distinct geometric shapes which vibrate at the same characteristic frequencies.'' In this presentation, Prof. Gordon will explain this result with the aid of some ``isospectral music'' prepared by Dennis DeTurck, and a film made by Jean-Pierre Bourguignon et al. For a popular account of this topic see Gordan and Webb's article in American Scientist (Jan.-Feb. 1996) or Ivars Peterson's recent piece here in MAA Online. Toby Driscoll has constructed animations of sound-alike drums vibrating, and S. Sridhar has come up with microwave cavities in the isospectral drum shapes which provide experimental verification of Gordon, Webb and Wolpert's mathematical result. Peter Buser, John H. Conway, Peter Doyle and Klaus-Dieter Semmler have constructed a whole ''gallery'' of sound-alike drums.

Herbert Edelsbrunner, Univ. of Illinois at Urbana-Champagne, will speak on "Computing Organic Shapes on a Micro and a Macro Scale"; Prof. Edelsbrunner works in computational geometry, and his recent work focuses on geometric shapes and their topological properties thus expanding computational geometry into what might be called computational topology. Topological questions of shape arise, for example, in applications of geometry to molecular biology, which will be the focus of this talk. Geometric models of proteins will be discussed and how they relate to computing volume and surface area, voids and pockets, symmetry and non-symmetry, docking between proteins and ligand, and deformations of proteins.

Walter Whiteley, York University, will address "Old Questions - New Answers: Geometry for Computer Aided Design". Says Prof. Whiteley: "At the core of many applications of mathematics lies the need to describe and predict the shape, form, and static and kinematic behaviour of discrete structures. Underlying these developments is an exciting blend of combinatorial, symbolic, and numerical methods, organized around a deep understanding of the classical and modern geometries. Applications in our (approximately) Euclidean world require methods from across the range of geometries: topological, combinatorial, projective, affine, Euclidean. Statics, local kinematics, computer vision, and multivariate splines all present a common core of combinatorial and projective geometric patterns, while the methods and unsolved problems of plane parametric CAD (Computer Aided Design) run the full gamut of geometries."

Delle Maxwell, Silicon Graphics, will speak on "Sharing the Mind's Eye: Computer Animation in Visualizing Geometry". Over the last few years, the Geometry Center, has broken new ground in the art of communicating mathematics by making several high quality videos to help communicate sophisticated ideas and discoveries in geometry to young audiences. Delle Maxwell, who served as graphic designer/animator and co-director for these landmark videos, will show highlights from the award winning Outside In (1994) concerning Thurston's beautiful sphere eversion, and Not Knot (1991) which explains the recently discovered connections between knots and the hyperbolic geometry associated with their complements. She will also talk about the mathematical/design/programming collaborations and challenges involved in communicating geometry through the medium of video.

Jeffrey Weeks, of Canton, NY, will wrap up the morning session by considering "The Shape of Space". This will bring together ideas explored in his book of the same name, and the new Geometry Center video created by Weeks, Maxwell and their colleagues (now available from Minnesota Council of Teachers of Mathematics). In particular, he will address a vital question which has puzzled philosophers and astronomers down through the ages: Is the universe finite or infinite? Dr. Weeks: "That ancient question may finally be answered using observational data to be available in the year 2001. This presentation will show how a universe can be finite, yet have no boundary. Freely available java games (used in grades 5-12) will introduce the basic concept. The Shape of Space video will then take the viewer on a computer-animated tour of several possible shapes for the universe. The presentation will conclude with an explanation of how the upcoming measurements of the cosmic microwave background radiation may be used to determine the exact shape of the real universe, assuming it's small enough that we can see at least `half way around'." Dr. Weeks is a pioneer in the design and development of geometry games as a tool for getting students to experience the geometry of manifolds without having to learn the theory first.

The three-hour afternoon session, ``Long Live Geometry!'', focusses on vital pedagogical issues. Here, the presenters will share their visions of how geometry should be taught in the coming decades, and why.

David Henderson, Cornell University, will open the proceedings with "Opening Students' Minds: Experiencing Non-Axiomatic Geometry in the Classroom"; this will reflect to a great extent the philosophy and methods of the presenter's books Experiencing Geometry on Plane and Sphere (Prentice Hall, 1996), and Differential Geometry: A Geometric Introduction (Prentice Hall, 1998), which provide the framework for hands-on, cooperative-learning based explorations of geometry, setting aside the restrictions of formalism. This distinctive approach stimulates and challenges students to develop a deeper understanding of mathematics through participation and through starting from intuitive understanding. Prof. Henderson, who has been teaching geometry this way for over twenty years, will share why he believes that most semesters he learns new geometry from about 40% of his students, especially, students who are different from him in gender, race, and cultural background (see his paper ``I Learn Mathematics From My Students - Multiculturalism In Action,'' in For The Learning of Mathematics, v16, 1996).

James King, Univ. of Washington, will follow by discussing "Reshaping School Geometry with Models and Software"; says Dr. King: "Mathematics teachers are living in the midst of currents of change and reform. What are appropriate methods for teaching geometry in the schools? What should geometry in the schools be? For seven years, the teachers at the IAS/Park City Mathematics Institute High School Teachers Program have wrestled with these questions while in the midst of an intense mathematics research environment. This report will describe examples of what they do in their schools, using models, projects, software, paper-folding, and balls, but also using ways of seeing mathematics and geometry everywhere."

John H. Conway, Princeton University, will speak on "Organizing the Geometry of the Triangle" in which he will share his vision for a modern-day approach to high school geometry. Says Prof. Conway, "A few decades ago there was a great revolt against the teaching of projective geometry in the Universities and Colleges, and unfortunately this was later followed by a revolt against the teaching of geometry in the high schools. In my view this has had a disastrous effect on mathematical education, because even people who are unhappy with formula can appreciate geometrical results visually. In Thurston's words, "Geometry is the user interface of Mathematics". Fortunately the situation is improving. High school students who a few years ago learned only a minuscule amount of geometry are now keenly using the excellent geometrical computer systems, and want to know more. We are about to experience a great rival of the "old" geometry, and in particular, of the classical geometry of the triangle. I have long been interested in triangle geometry, and being a professional mathematician, am keen to regard it as a coherent theory rather than a collection of individually interesting theorems. In my talk, I shall describe some famous theorems about triangles and some less famous ones, and will attempt to describe the many links between them.

Joseph Malkevitch, York College (CUNY), will discuss "New Geometry Helps Emerging Technologies from Robotics to HDTV"; says Prof. Malkevitch: "Geometry was born in an attempt to understand physical space. However, recently geometry has grown to encompass visual phenomena in a broad sense. A surprising consequence of this expansion in geometry's domain has been widespread application of geometry to a variety of areas associated with rapidly emerging new technologies such as medical imaging, robotics, compact disks, fax, wireless telephony, and high definition television. Since many of these new geometric discoveries can be taught in a way which builds on intuitively appealing ideas, the connection between these new geometric results and new technologies can be taught in grades K-12."

Finally, co-organizer Colm Mulcahy, Spelman College, will try to put all of this in perspective with "Exploring Frontiers in Geometry - Looking Back and Looking Forward". Says Dr. Mulcahy, "Geometry, with its intuitive pictorial appeal, is arguably the branch of mathematics most suited to communication to a general audience. Yet for many, including some with advanced mathematical training, the ancient roots of geometry can lead to a mistaken impression that while the subject is certainly elegant, it is essentially static. In this brief final presentation, I will attempt to dust off any remaining cobwebs perceived to be clinging to this field, and try to mould the diverse viewpoints expressed in the earlier talks into a unified picture of geometry as a thriving, dynamic endeavour which is also in need of fresh talent and ideas to keep it alive and kicking well into the next century."

Of related interest, Robert Osserman, Stanford University & MSRI, is organizing a symposium entitled "From Riemann to Strings: The Ongoing Romance between Geometry and Physics", on the afternoon of Monday, Feb 16.

The entire program for the meeting is available online.

MAA Online is edited by Fernando Q. Gouvêa
Last modified: Sat Oct 25 09:46:20 1997 Example: David Eppstein`s Geometry in Action and Geometry Junkyard.